Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Physics and Astronomy

Committee Member

Dr. Murray S. Daw, Committee Chair

Committee Member

Dr. Catalina Marinescu

Committee Member

Dr. Jian He

Committee Member

Dr. Jens Oberheide


Thermal properties of solids are one of the most important features studied in solid state physics, and many of the thermal properties are consequences of anharmonicity in solids. Many experimental techniques and theoretical methods have been successfully developed in the study of the anharmonicity and our understanding of the anharmonicity of vibrational modes in materials is emerging nicely. A new theoretical approach that is based on moments of the Liouvillian has been developed by Dr. Murray Daw's group. The new approach is expected to be much faster than other theoretical methods. Furthermore, the new approach does not involve perturbation theory and is able to overcome specific difficulties and limitations of other methods. My work presented in the thesis mainly involves two aspects: 1, I have helped with the implementation based on semi-empirical potentials; 2, I have studied various kinds of nano-materials using the new approach. For graphene, we study how the anharmonicity (measured by temperature) affects the flexural modes. We find that the dispersion relation of the flexural modes of free-standing graphene is renormalized by anharmonic coupling to other modes. It has been argued in the literature that such anharmonic coupling keeps the graphene sheet stable and relatively flat. Our results agree with anharmonic continuum theory at low temperatures and for small wave vectors, where the continuum theory is supposed to be correct. Not limited to perturbative treatment, our work extends the results to higher temperature and larger wave vector. Our results are not accessible by experiments because the experiments on graphene is too difficult for current equipments. In the method we assume that the average of the amplitude of any vibrational mode equals zero. The assumption holds for all modes except some specific modes. To account for this special case, we have had to make modifications to the theory, as explained in Chapter 5, then to the codes for the calculations for carbon nanotubes and fullerenes. For carbon nanotubes, we study the anharmonic frequencies of all vibrational modes including the radial breathing mode. We demonstrate that the anharmonicity (defined in terms of normalized frequency) is a linear function of temperature. We show that the vibrational modes can be categorized into 3 groups in terms of their temperature dependence and the radial breathing mode is the most anharmonic among all modes. We have treated various nanotubes with different chiralities, lengths and diameters and have found the tubes with larger diameters show more anharmonicity. Our results agree with more limited calculations based on molecular dynamics and available experiments. For fullerenes, we find that the frequencies of all vibrational modes drop systematically with temperature. As far as we know, there are no previous calculations and our results fill in a two-decade gap in the theoretical treatment of the anharmonicity of fullerenes. Our results agree with the experimental results (for limited number of modes present in the experiments). Our results provide clearer pictures on the anharmonicity of vibrational modes in the carbon nano-materials.